The point masses m and 2m lie along the x-axis, with m at the origin and 2m at x = L. A third point mass M is moved along the x-axis. (a) At what point is the net gravitational force on M due to the other two masses equal to zero?

Verified step by step guidance

1. Identify the forces acting on mass M due to the gravitational attraction from masses m and 2m. Use Newton's Law of Universal Gravitation, which states that the gravitational force between two masses is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force exerted by m on M is F1 = GmM/x^2, and the force exerted by 2m on M is F2 = G(2m)M/(L-x)^2, where G is the gravitational constant.
2. Set up the equation for the net force acting on mass M by considering the direction of the forces. Since m is at the origin and 2m is at L, if M is between m and 2m, the forces due to m and 2m are in opposite directions. The net force F_net = F1 - F2 when M is between m and 2m.
3. Simplify the equation by canceling out common terms. The mass M and the gravitational constant G can be canceled out from the equation, leading to m/x^2 = 2m/(L-x)^2.
4. Rearrange the equation to find x, the position of M where the net force is zero. This can be done by cross-multiplying and simplifying the equation to find a quadratic equation in terms of x.
5. Solve the quadratic equation obtained in the previous step to find the value of x. This value will give the position along the x-axis where the net gravitational force on M due to m and 2m is zero.

Key Concepts

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force (F) is proportional to the product of the masses (m1 and m2) and inversely proportional to the square of the distance (r) between them, expressed as F = G(m1*m2)/r^2, where G is the gravitational constant. Understanding this force is crucial for analyzing the interactions between the point masses in the problem.

Equilibrium Condition

The equilibrium condition in this context refers to the state where the net gravitational force acting on the mass M is zero. This occurs when the gravitational forces exerted by the two other masses (m and 2m) on M are equal in magnitude but opposite in direction. To find the position of M where this condition holds, one must set the forces from both masses equal to each other and solve for the position along the x-axis.

Coordinate System

A coordinate system provides a framework for locating points in space, which is essential for solving problems in physics. In this scenario, the x-axis is used to define the positions of the masses: m is at the origin (0,0) and 2m is at (L,0). By establishing a clear coordinate system, one can effectively analyze the distances and forces acting on the mass M as it moves along the x-axis.